The short answer is, no, they don't stick. The only time you see lightsaber wielders actually stop an opposing lightsaber is when one person is holding two sabers in an X formation. From the various sources of what I've read, the closest physical-feeling equivalent would be fencing with PVC pipes - they don't stick, and in fact bounce a little as the energy blades repulse each other slightly.
According to this highly specifically-titled book there are 7 different styles of lightsaber combat. The act of cutting off the opponent's sword or blaster hand is referred to as Cho Mai. While we look at the removal of limbs as grotesque or horrible, the Jedi looked at it as honorable, as you're removing the ability for the opponent to fight without killing them and causing minimal physical damage (I'm sure that the advances in medical technology help that worldview); conversely, the Sith view it as merciful and a symbol of power as you're bringing the target low, but actively choosing to not kill them outright.
Although I don't have any in-universe data to back it up, I'd also say that the training for lightsaber use would be the main reason that it isn't as much of a problem. The weapon doesn't lend itself to close quarters fighting (although there is one form of the seven devoted to it), but the full style of fighting is a fluid one, and attempting to "slide down the blade" would be fruitless as the other wielder would just keep moving, presumably in some sort of parry. Basically the only time you ever see a wielder holding the blade steady for an extended period is when Qui'Gon is punching holes in doors.
Obviously the commonplace removal of limbs in lightsaber combat is the one of, if not the main, reason Kylo Ren decided to use the crossguard in his lightsaber in Episode 7, but that remains to be seen.